Integrable generalizations of oscillator and Coulomb systems via action–angle variables

Abstract

Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N−1)-dimensional system. We present the action–angle formulation of such models in terms of the radial degree of freedom and the action–angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.